Banach subalgebra generated by one element

For $x$ in a complex unital Banach algebra $A$, the closed unital subalgebra it generates is $B=\overline{\mathbb{C}[1,x]}$. Its character space is identified with ${\sigma }_B(x)$ by $\phi \mapsto \phi (x)$: polynomial density forces uniqueness, and spectrum equals character values in a commutative Banach algebra gives existence.

The ambient algebra matters. Spectrum of a polynomial-generated Banach subalgebra shows that ${\sigma }_B(x)$ is the full spectrum of $x$ in $A$, including all bounded complementary components of ${\sigma }_A(x)$. For example, in $A=C(\mathbb{T})$ the coordinate $x(z)=z$ has spectrum the unit circle, whereas $B$ is the boundary realization of the disk algebra and ${\sigma }_B(x)$ is the closed unit disk. Consequently it is the complement of ${\sigma }_B(x)$, rather than necessarily that of ${\sigma }_A(x)$, which is connected.

For $B=\overline{\mathbb{C}[1,x]}$ inside $A$, the displayed set is the full spectrum of $x$ in $A$. On a bounded complementary component, the maximum modulus principle gives $|p(\lambda )|\le {\max }_{{\sigma }_A(x)}|p|\le \Vert p(x)\Vert$. Evaluation $p(x)\mapsto p(\lambda )$ therefore extends to a character of an algebra on $B$, showing $\lambda \in {\sigma }_B(x)$. Conversely the unbounded resolvent component survives by resolvent components in a closed unital subalgebra. In particular, if zero lies in that component, $x^{-1}$ belongs to the norm closure of polynomials in $x$.